Theta-quarks

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From the course by National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

11 ratings

National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

11 ratings

This is a master course given in Moscow at the Laboratory of Algebraic Geometry of the National Research University Higher School of Economics by Valery Gritsenko, a professor of University Lille 1, France. Jacobi forms are holomorphic functions in two complex variables. They are modular in one variable and abelian (or double periodic) in another variable. The theory of Jacobi modular forms became an independent research subject after the famous book of Martin Eichler and Don Zagier “Jacobi modular forms” (Progress in Mathematics, vol. 55, 1985) which was cited more than a thousand times in research papers. This is due to many applications of Jacobi forms in arithmetic, topology, algebraic and differential geometry, mathematical and theoretical physics, in the theory of Lie algebras, etc. The list of mentioned subjects shows that my course might be useful for master and Ph.D. students working in different directions. Motivated undergraduate students can also study this subject. To follow the course one has to know only elementary basic facts from the theory of modular forms (for example, the paragraphs 1-4 of the chapter VII of Serre’s “A Course in Arithmetic” are enough). The main hero of the course is the Jacobi theta-series. Using it we will construct a lot of concrete examples of Jacobi forms in one or many abelian variables, in particular, Jacobi forms for root systems. For some of you, who will be successful with the theoretical exercises of the course, I am ready to formulate research problems for Master or Ph.D. thesis. (Ph.D. support might be available at CEMPI in Lille or at the Faculty of Mathematics of National Research University Higher School of Economics in Moscow)

From the lesson

Theta-blocks, theta-quarks and the first Jacobi cusp form of weight 2

This module is devoted to very important notion of theta-blocks and theta-quarks. In this module we also construct the first Jacobi form of weight 2. Also there is a peer review in the end of this module.

Theta-blocks14:49

Theta-blocks (part 2)13:59

Theta-quarks14:06

Dimension of the space of Jacobi forms of odd weight14:43

Theta-quarks (part 2)11:41

The proof of theorem about theta-quarks16:11

The first Jacobi cusp form of weight 210:48

Meet the Instructors

Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

[MUSIC]

The question which we would like to solve now

How to construct

cusp product

Theta a1.

So, theta a8.

Cusp being the correspondent, Jacobi modular form is a cusp form.

Proposition

Let a and b be positive.

Then the product of two thetas,

theta a x theta b.

The Jacobi form of weight 1 index

a to the square plus b to the square over 2 with respect

to the character not multiply system (v eta to the power 6 x

v H to the power a + b).

This product is a cusp

If and only if the number

a times b over the square of the greatest

common divisor a, b is even.

The proof is rather simple.

Let us calculate Ze for the expansion of this product.

With a respect to the 2 in this instance.

The connecting symbol is a character.

Q to the power n1 to the square

+ n2 to the square / 8, r to the power.

an1 + bn2 / 2.

As usual, r = e to the power 2 pi i z,

and q is e to the power 2 pi i tau.

So this is our 4e and k efficient.

Certainly we can make a re-summation to fix the same

N/2 and

the same, l/2.

Then the hyperbolic norm.

Over the index for Norm 0 Fourier coefficient.

Sorry here not n over 2 but

This is n over 8 the node of

the corresponding index, and

the index here, is the first part.

The 5th called into this index n over 2 in the second l over 2,

this is the index of non-0 of Fourier coefficient Is equal,

4 times the index

of Jacobi form * N /8.

L /2, to the square.

This our usual hyperbolic norm for

nm minus n2 square.

In this particular case

we have 1/4 because

4/16 is 4 x N1 to the square

plus N2 to the square

x a square + b square-

a plus 1 plus b squared.

Let me correct this formula a little bit.

And this is certainly equal.

To 1/4(n1b-

N2a) to the square.

Please check this.

It's very simple calculation.

And certainly, this known is greater or equal to 0.

And this inequality reflects the product of two holomorphic Jacobi forms.

But now let's assume that a and b are.

This gives us no restriction at all.

Then, a x b is even.

But, n1 and n2,

they both, are odd.

Because we have this character in the Fourier summation.

So the Kronecker symbol is 0 if the product n1 x n2 is even.

But now, a or b is

even, and n1 and n2 are odd.

So we see that this square is strictly positive.

Due to these two It

gives us the proof of the proposition.

So if the product of two jacobi is a cusp form under this condition.

And now we can formulate

the correspondent result

about the product of 8 or

more generally kappa

theta series corollary.

I would like to use the following notation.

That A is the vector,

A 1 A, K.

All A, E, are strictly positive.

Then theta with index the vector

a is by definition the product

of k theta functions.

Let's for simplicity, there is not really friction.

We, assume

that the greatest common divisors,

maybe I write it explicit form.

The greatest common divisor

of a1, ak, is equal to 1.

Then.

Theta a Is a cusp Jacobi form

If At least

One ai is even.

Moreover, this function, theta a,

is a Jacobi form of weight K/2

of index A1 to the square

AK to the square/2 with

trivial character.

If and only if

K is congruous to 0,

modular 8,

then the sum aK is even and

under this condition this

Jacobi form is a cusp form.

If and only if At

least one a:e is even.

So we solve the problem.

We describe, all theta product.

We check as form.

And I can construct the simplest examples.

We take, with a vector a, 6 units.

And 2, 2.

There's the correspondent.

Data function.

Eta to the power 6 x theta 2 to

the square is jacobi form of J4 and

this is 4 and

index 6 + 4 + 4/2.

And index 7.

This is the first jacobi cusp form which we can construct using the theta product.

[MUSIC]

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